1.

Show that (i) `x+3 ` is factor of `69+11x-x^(2)+x^(3)` (ii) `2x-3` is factor of `x+2x^(3)-9x^(2)+12`

Answer» (I) let `p(x) =x^(3)-x^(2)+11 x+69`
we have to show that , x+3 is a factor of p(x) .
`i.e., p(-3)=0`
Now `p(-3)=(-3)^(3)-(-3)^(2)+11(-3)+69`
`=-2-9-33+69=-69+69=0`
hence ,(x+3) is a factor of p(x) .
(ii) Let `p(x) =2x^(3)-9x^(2)+x+12`
we have to show that ,2x -3 is a factor of p(x).
i.e., `p((3)/(2))=0`
Now ,
`p((3)/(2))=2((3)/(2))^(3)-9((3)/(2))^(2)+(3)/(2)+12`
`=2xx(27)/(8) -9xx(9)/(4)+(3)/(2)+12`
` =(27)/(4)-(81)/(4) +(3)/(2)+12`
`=(27-81+6+48)/(4)=(81-81)/(4)=0`
Hence ,`(2x-3)` is a factor of p(x).


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